I actually work as a flight nurse for an international air ambulance company, as part of the morning pre flight brief, diversions are always discussed. The usual plan is that we divert to the nearest biggest airport as they normally have hospitals in the area capable of dealing with most emergencies. This means we will possibly pass smaller airports.
A flight from anywhere in Europe, a major airport is usually reachable quite easily, from the Canary Islands which are a frequent run, after departure if closer to land and carrying on is a shorter time, you have as you come north places like Lisbon, Porto etc. Even across Asia the same generally applies.
Africa is a case all on its own for medical facilities and the option if stability is achieved is to continue until you can get to a facility that has the required equipment. Nairobi is about the best, followed by Cairo.